Steepest Decent and Conjugate Gradients (CG)

Solving of the linear equation system

Steepest Decent and Conjugate Gradients (CG) Solving of the linear equation system Problem: dimension n too big, or not enough time for gauss elimination Iterative methods are used to get an approximate solution.

Steepest Decent and Conjugate Gradients (CG) Solving of the linear equation system Problem: dimension n too big, or not enough time for gauss elimination Iterative methods are used to get an approximate solution. Definition Iterative method: given starting point, do steps hopefully converge to the right solution

starting issues

Solving is equivalent to minimizing

starting issues Solving is equivalent to minimizing A has to be symmetric positive definite:

starting issues

starting issues If A is also positive definite the solution of is the minimum

starting issues If A is also positive definite the solution of is the minimum

starting issues error: The norm of the error shows how far we are away from the exact solution, but can’t be computed without knowing of the exact solution.

starting issues error: The norm of the error shows how far we are away from the exact solution, but can’t be computed without knowing of the exact solution. residual: can be calculated

Steepest Decent

We are at the point. How do we reach ?

Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( )

Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go?

Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

Steepest Decent one step of steepest decent can be calculated as follows:

Steepest Decent one step of steepest decent can be calculated as follows: stopping criterion: or with an given small It would be better to use the error instead of the residual, but you can’t calculate the error.

Steepest Decent Method of steepest decent:

Steepest Decent As you can see the starting point is important!

Steepest Decent As you can see the starting point is important! When you know anything about the solution use it to guess a good starting point. Otherwise you can choose a starting point you want e.g..

Steepest Decent - Convergence

Definition energy norm:

Steepest Decent - Convergence Definition energy norm: Definition condition: ( is the largest and the smallest eigenvalue of A)

Steepest Decent - Convergence Definition energy norm: Definition condition: ( is the largest and the smallest eigenvalue of A) convergence gets worse when the condition gets larger

Conjugate Gradients

is there a better direction?

Conjugate Gradients is there a better direction? Idea: orthogonal search directions

Conjugate Gradients is there a better direction? Idea: orthogonal search directions

Conjugate Gradients is there a better direction? Idea: orthogonal search directions only walk once in each direction and minimize

Conjugate Gradients is there a better direction? Idea: orthogonal search directions only walk once in each direction and minimize maximal n steps are needed to reach the exact solution

Conjugate Gradients is there a better direction? Idea: orthogonal search directions only walk once in each direction and minimize maximal n steps are needed to reach the exact solution has to be orthogonal to

Conjugate Gradients example with the coordinate axes as orthogonal search directions:

Conjugate Gradients example with the coordinate axes as orthogonal search directions: Problem: can’t be computed because (you don’t know !)

Conjugate Gradients new idea: A-orthogonal

Conjugate Gradients new idea: A-orthogonal Definition A-orthogonal: A-orthogonal (reminder: orthogonal: )

Conjugate Gradients new idea: A-orthogonal Definition A-orthogonal: A-orthogonal (reminder: orthogonal: ) now has to be A-orthogonal to

Conjugate Gradients new idea: A-orthogonal Definition A-orthogonal: A-orthogonal (reminder: orthogonal: ) now has to be A-orthogonal to

Conjugate Gradients new idea: A-orthogonal Definition A-orthogonal: A-orthogonal (reminder: orthogonal: ) now has to be A-orthogonal to can be computed!

Conjugate Gradients A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram- Schmidt (same idea as Gram-Schmidt).

Conjugate Gradients Gram-Schmidt: linearly independent vectors

Conjugate Gradients Gram-Schmidt: linearly independent vectors

Conjugate Gradients Gram-Schmidt: linearly independent vectors conjugate Gram-Schmidt:

Conjugate Gradients A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram- Schmidt (same idea as Gram-Schmidt). CG works by setting (makes conjugate Gram- Schmidt easy)

Conjugate Gradients A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram- Schmidt (same idea as Gram-Schmidt). CG works by setting (makes conjugate Gram- Schmidt easy) with

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Conjugate Gradients

Method of Conjugate Gradients:

Conjugate Gradients - Convergence

Conjugate Gradients - Convergence for steepest decent for CG Convergence of CG is much better!